TRANSFORM CODING. y = F x. y q = Q[y] 4. F is chosen so that many elements of y will have small amplitudes.

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1 SIGNAL COMPRESSION. Lossless (a) invertible (b) compression ratio: - 3 (c) Huffman coding (d) Ziv-Lempel algorithm. Lossy (a) non-invertible (b) compression ration: 5-5 (c) JPEG (Image compression) (d) LPC (Speech compression) (e) MP3 (Audio compression)

2 TRANSFORM CODING. Transform. Quantize y = F x y q = Q[y]. Frequently, F is an orthonormal matrix.. F should decorrelate the elements of x. 3. The elements of y will have different amplitudes.. F is chosen so that many elements of y will have small amplitudes. 5. Usually an efficiently implemented F is chosen. (So that y = F x can be performed with less than N operations.) TRANSFORM DECODING. Inverse Transform r = F y q

3 TRANSFORM DECODING If F is orthonormal, then y(k) = y t y k = (F x) t (F x) = x t F t F x = x t I x = x t x = x(n) n So the energy (sum of squares) is the same in the signal domain as in the transform domain. (This is a general form of Parseval s theorem.) Also: (y q (n) y(n)) = (y q y) t (y q y) k = (F r F x) t (F r F x) = (r x) t F t F (r x) = (r x) t (r x) = (r(n) x(n)) n So the distortion can be computed in either the signal domain or in the transform domain. The distortion is defined here as E = n (r(n) x(n)). 3

4 TRANSFORM CODING Different transforms can be used.. DCT (Discrete Cosine Transform) - JPEG. DWT (Discrete Wavelet Transform) - JPEG 3. Filter banks - MP3 Audio

5 DISCRETE COSINE TRANSFORM (DCT). The DCT is like a real-valued version of the DFT.. There are types of DCT. (DCT-I, DCT-II, DCT-III, DCT-IV) 3. Each DCT is an orthonormal transform. (F t F = I.). The DCT-II is the most commonly used type. 5. The element y(k) is an inner product of x with the kth row of F. y(k) = N n= f(k, n) x(n) 6. The DCT gives a real-valued frequency decomposition of a real-signal x(n). 7. The DCT can be derived from the DFT by symmetric extension. 8. Different types of symmetric extension give rise to the four types of DCT. 9. The DCT can be computed using the DFT (or FFT). The DCT-II is given by where y(k) = x(n) = N N N n= N k= ( ) πk b(k) x(n) cos (n +.5) N ( ) πk b(k) y(k) cos (n +.5) N b() = /, b(k) =, k N The signal x(n) is given by a sum of cosine functions, so the DCT is a discrete-time version of the Fourier series using only cosine. 5

6 DISCRETE COSINE TRANSFORM (DCT) The basis functions f(k, n) of the 6-point DCT-II

7 DISCRETE COSINE TRANSFORM (DCT) The spectrum of each of the 6-point DCT-II basis functions

8 DISCRETE COSINE TRANSFORM (DCT) As a simple illustration of Transform Coding with the DCT we can use a very simple quantization scheme - set the smallest M coefficients to zero. Here we set 5 (out of 5) coefficients to zero. 5 SIGNAL x(n) n DCT COEFFICIENTS k SET M DCT COEFFICIENTS TO ZERO k RECONSTRUCTED SIGNAL n 8

9 DISCRETE COSINE TRANSFORM (DCT) To set the smallest M DCT coefficients to zero we can use the following Matlab code. % Compute DCT of Signal y = dct(x); % Find indices of smallest coefficients [y_sorted, k] = sort(abs(y)); % Set M smallest coefficients to zero M = 5; yq = y; yq(k(:m)) = ; % Compute Inverse DCT r = idct(yq); 9

10 DISCRETE COSINE TRANSFORM (DCT) The error between the original signal and the reconstructed signal is very small: 5 ERROR SIGNAL From the plot of the DCT coefficients we could predict that the error is small because the smallest 5 coefficients are very small in magnitude compared with the remaining 5 coefficients. We can make a plot of the distortion (square error) as a function of the number of coefficients set to zero (M). ERROR AS A FUNCTION OF M (#COEFFS SET TO ZERO) M 6 log(error) AS A FUNCTION OF M (#COEFFS SET TO ZERO) M

11 TWO-DIMENSIONAL DCT. For image compression (as in JPEG) we need to take the DCT of a D array.. JPEG stands for Joint Photographic Experts Group. 3. The D DCT of a D array is performed by computing the DCT of each row of the array, followed by computing the DCT of each column of the resulting D array.. The D DCT represents a D array as a sum of D cosine functions. We can view these D basis functions as we viewed the D basis functions of the D DCT. The D DCT can be implemented in Matlab using the commands % x is a D array >> x = rand(6); % Compute the D DCT of x >> y = dct(dct(x). ). ; % dct(x) computes the DCT of each column of x % dct(x. ). compute the DCT of each row of x % Compute the D Inverse DCT of y >> r = idct(idct(y). ). ; % Verify that r == x >> e = r - x; max(abs(e(:))) 5.55e-6

12 TWO-DIMENSIONAL DCT The first 3 of the 6 basis functions of the 8 by 8 point D DCT.

13 TWO-DIMENSIONAL DCT The second 3 of the 6 basis functions of the 8 by 8 point D DCT. 3

14 TWO-DIMENSIONAL DCT. In practice (eg: JPEG), an image is segmented into blocks (8 by 8 pixels, or 6 by 6 pixels) and the D DCT is applied to each block.. To illustrate the effect of DCT-based compression, lets set the smallest 8 DCT coefficients of each 8 by 8 block to zero. ORIGINAL IMAGE SETTING 3/ THE DCT COEFFS TO ZERO There is almost no noticeable difference!

15 TWO-DIMENSIONAL DCT If we set 6 DCT coefficients to zero out of every 6-coefficient block, we get the following result. ORIGINAL IMAGE SETTING 6/6 OF THE DCT COEFFS TO ZERO This is a compression ration of 6/6 = 93%. 5

Image Compression through DCT and Huffman Coding Technique

Image Compression through DCT and Huffman Coding Technique International Journal of Current Engineering and Technology E-ISSN 2277 4106, P-ISSN 2347 5161 2015 INPRESSCO, All Rights Reserved Available at http://inpressco.com/category/ijcet Research Article Rahul